Re: What databases have taught me

Marshall <marshall.spight_at_gmail.com> wrote:
> > - You don't need faces; just a (unordered) nodes and edges.
>
> Are you sure?
>
> Consider a square, with four points, top-left, top-right, bottom-left,
> bottom-right. Now add a fifth edge, from top-right to bottom-left.
>
> If this edge goes through the interior of the square, that is a
> different set of regions than if it goes around the outside of
> the square.

Yes, those are different planar embeddings, so they have different
duals. (The duals happen to be isomorphic to each other in this case,
but that's just a coincidence; not necessarily guaranteed to always be
true.) This gets back to the fact that duals are really an operation on
planar embeddings[*] of graphs rather than on graphs. The phrase "dual
of a graph" means "a dual of some planar embedding of that graph". So
using either embedding would give you a correct answer. However, yes
you would need to pick an embedding of the graph at some point to use
the geometric algorithm. (The combinatorial dual, which is equivalent
to a geometric dual but is defined in a completely different way, may be
more amenable to computation via the relational model than a planar
embedding of the graph. However, you'll have to find someone who knows
about it to explain combinatorial duals to you.)

I didn't think of that as needing faces... mainly because the word
"faces" was used in a different sense by the MathWorld article that
someone posted a link to earlier (it was talking about the threedimensional
analog of a dual for a polyhedral graph), and I just assumed
that you saw the word "faces" there.

[*] Since someone brought up topology, I suppose this would be an
opportune time to point out that duals are formally described in terms
of embeddings on a sphere, rather than a plane. That turns out to be
equivalent, because the embedding can't possibly depend on every single
point of the sphere, so it's always possible to remove a point of the
sphere and thus turn it into the topological equivalent of a plane. The
use of a sphere turns out to be different from use of a plane, except
that it removes that messiness about having to specify that "outside"
counts as a region. I am not familiar with embeddings on torii (sp?)
and thus can't comment on that aspect of the conversation. It certainly
seems reasonable that there could be graphs that could be embedded on
torii but which are not planar, in which case there may be something
like a torus-embedding that's analogous to a dual; but if so, that is
not what is normally meant by the word "dual".